Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C7×C64 ⋊ C2 |
| Sylow | C7 ⋊ SD128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D224 |
| commutator, cosocle | C7×C32 | C22 |
| Frattini | C32 | C2×D7 |
| Fitting | C7×C64 | C2 |
| socle | C2×C7 | D32 |
| prime radical | D224 | C2 |
| hypercenter | C64 | D7 |
| hypocenter | C7 | SD128 |
| LCS | G ▷ C7×C32 ▷ C7×C16 ▷ C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 71 |
| transitive degree | μt(G) = 448 |
Polycyclic presentation:
2^7,7|a^2=g,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^6,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^6227 conjugacy classes — view the full character table.
| maximal subgroup of | D7×SD128, C2×#11725240, #13274849, … |
| maximal normal subgroup of | D7×SD128, C2×#11725240, #13274849, … |
| maximal quotient of | #13241675, #13241678, #13241698, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 6);